Given the function f(x)=sin2x+23sin(x+4π)cos(x−4π)−cos2x−3. (1) Find the interval(s) where the function f(x) is monotonically decreasing. (2) Find the maximum value of the function f(x) on the interval [−12π,3625π].
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
Solution
(1) The function f(x)=sin2x+23sin(x+4π)cos(x−4π)−cos2x−3 can be simplified as follows: f(x)=−cos2x+23⋅(22sinx+22cosx)⋅(22cosx+22sinx)−3 f(x)=−cos2x+23⋅(21+21sin2x)−3 f(x)=3sin2x−cos2x=2sin(2x−6π) Let 2kπ+2π⩽2x−6π⩽2kπ+23π, we get kπ+3π⩽x⩽kπ+65π. Thus, the intervals where the function is decreasing are [kπ+3π,kπ+65π], where k∈Z.
(2) On the interval [−12π,3625π], we have 2x−6π∈[−3π,911π]. The maximum value of f(x) occurs when 2x−6π=2π. Therefore, the maximum value of the function f(x) is 2.
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